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1. Boundary unique continuation for the Laplace equation and the biharmonic operator

   S. Berhanu (January 2022, Communications in analysis and geometry)

   Kefeng Liu (Ed.)

   We establish results on unique continuation at the boundary for the solutions of ∆u = f, f harmonic, and the biharmonic equation ∆^2u = 0. The work is motivated by analogous results proved for harmonic functions by X. Huang et al in [HK1], [HK2], and [HKMP] and by M. S. Baouendi and L. P. Rothschild in [BR1] and [BR2]. 

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2. A Highly Efficient Dual-band Harmonic-tuned GaN RF Synchronous Rectifier with Integrated Coupler and Phase Shifter

   https://doi.org/10.1109/MWSYM.2019.8700900  

   Hoque, Md Aminul; Ali, Sheikh Nijam; Mokri, Mohammad Ali; Gopal, Srinivasan; Chahardori, Mohammad; Heo, Deukhyoun (June 2019, 2019 IEEE MTT-S International Microwave Symposium (IMS))

   This paper presents a dual-band RF rectifying circuit for wireless power transmission at 1.17 GHz and 2.4 GHz. A dual-band harmonic-tuned inverse-class F/class-F mode power amplifier using a 10 W GaN device has been utilized to implement the proposed rectifier with an on-board coupler and phase shifter. The matching circuit is precisely designed so that the circuit operates in inverse class F and class F mode in the lower and upper frequency bands using dual-band harmonic tuning, respectively. Measurement results show that the rectifier circuit has 78% and 76% efficiencies at 1.17 GHz and 2.4 GHz frequency bands, respectively. To the best of the authors' knowledge, this rectifier is the first demonstration of a dual-band harmonic-tuned synchronous rectifier using a GaN HEMT device with an integrated coupler and phase-shifter for a watt-level RF input power. 

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3. Resistance scaling on 4 N -carpets

   https://doi.org/10.1515/forum-2020-0330  

   Canner, Claire; Hayes, Christopher; Huang, Robin; Orwin, Michael; Rogers, Luke G. (December 2021, Forum Mathematicum)

   Abstract The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F , let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass,On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n . Such estimates have implications for the existence and scaling properties of Brownian motion on F . 

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4. Classification of metric measure spaces and their ends using p-harmonic functions

   https://doi.org/10.54330/afm.120618  

   Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari (March 2022, Annales Fennici Mathematici)

   By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite \(p\)-energy \(p\)-harmonic and \(p\)-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local \(p\)-Poincaré inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We study the inclusions between these classes of metric measure spaces, and their relationship to the \(p\)-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant \(p\)-harmonic functions with finite \(p\)-energy as spaces having at least two well-separated \(p\)-hyperbolic sequences of sets towards infinity. We also show that every such space \(X\) has a function \(f \notin L^p(X) + \mathbf{R}\) with finite \(p\)-energy. 

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5. Chip-based self-referencing using integrated lithium niobate waveguides

   https://doi.org/10.1364/OPTICA.392363  

   Okawachi, Yoshitomo; Yu, Mengjie; Desiatov, Boris; Kim, Bok_Young; Hansson, Tobias; Lončar, Marko; Gaeta, Alexander_L (June 2020, Optica)

   The measurement and stabilization of the carrier–envelope offset frequency $f_{\mathrm{C}\mathrm{E}\mathrm{O}}$via self-referencing is paramount for optical frequency comb generation, which has revolutionized precision frequency metrology, spectroscopy, and optical clocks. Over the past decade, the development of chip-scale platforms has enabled compact integrated waveguides for supercontinuum generation. However, there is a critical need for an on-chip self-referencing system that is adaptive to different pump wavelengths, requires low pulse energy, and does not require complicated processing. Here, we demonstrate efficient $f_{\mathrm{C}\mathrm{E}\mathrm{O}}$stabilization of a modelocked laser with only 107 pJ of pulse energy via self-referencing in an integrated lithium niobate waveguide. We realize an $f\text{-}2f$interferometer through second-harmonic generation and subsequent supercontinuum generation in a single dispersion-engineered waveguide with a stabilization performance equivalent to a conventional off-chip module. The $f_{\mathrm{C}\mathrm{E}\mathrm{O}}$beatnote is measured over a pump wavelength range of 70 nm. We theoretically investigate our system using a single nonlinear envelope equation with contributions from both second- and third-order nonlinearities. Our modeling reveals rich ultrabroadband nonlinear dynamics and confirms that the initial second-harmonic generation followed by supercontinuum generation with the remaining pump is responsible for the generation of a strong $f_{\mathrm{C}\mathrm{E}\mathrm{O}}$signal as compared to a traditional $f\text{-}2f$interferometer. Our technology provides a highly simplified system that is robust, low in cost, and adaptable for precision metrology for use outside a research laboratory. 

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